The join construction for free involutions on spheres
Given a Poincaré complex X, the set of manifold structures on X x Dk relative to the boundary can be viewed as the k-th homotopy group of a space Ss(X). This space is called the block structure space of X. Free involutions on spheres are in one-to-one correspondence with manifold structures on real projective spaces. There is Wall’s join construction for free involutions on spheres which we generalize to a map between block structure spaces of real projective spaces. More precisely, we define a functor from the category of real finite-dimensional vector spaces with inner product to pointed spaces which to a vector space V assigns the block structure space of the projective space of V. We study this functor from the point of view of orthogonal calculus of functors, we prove that the 6-fold delooping of the first derivative spectrum of this functor is an W-spectrum. The proof uses mainly codimension 1 surgery theory. This results suggests an attractive description of the block structure space of the infinite-dimensional real projective space which is a colimit of block structure spaces of projective spaces of finite-dimensional real vector spaces. The description is via the Taylor tower of orthogonal calculus.